Circuits, Devices, Networks, and Microelectronics
CHAPTER 17. LINEAR ACTIVE FILTERS AND FREQUENCY PROFILES
17.1 INTRODUCTION
All circuits have a frequency response, characterized by their connected set of reactive components and
devices and their mix of intrinsic time constants. In many instances it is desired that an analysis of the
frequency character of a circuit be accomplished. But in other instances it is desired to adjust the circuit’s
frequency behavior and define or redefine the frequency profile of the circuit.
In most instances a frequency profiles is defined in terms of amplitude ‘pass’ functions. There are four
basic types:
(1)
(2)
(3)
(4)
low-pass
high-pass
band-pass
band-stop
Unless semiconductor components dominate the circuit, its time constants should be expected to relate to
capacitances and inductances and resistances. In this respect, intermediate frequency and RF (radiofrequency) profiles may be defined by pF or fF capacitances and H inductances. Profiles also may be
defined by RLC resonance networks and the placement of poles and zeros. High RF profiles usually need
to accommodate the effects of parasitic components such as leakage paths, wiring inductances, and
fringing capacitances. Frequency profiles in the UHF range (300MHz to 3GHz) or higher will relate to
microstrips, resonant cavities or artificial crystals for realizations of desired frequency responses.
At lower frequencies, typically associated with audio systems or other biological interface systems, many
components may be large and cumbersome. Consequently techniques have been developed in which
amplification loops are used to define profiles. These circuit topologies are usually called active filters.
Active filters are also used at higher frequencies in order to be able to deploy the function on an
integrated circuit, where all components will be small in both value and. size
An active filter is a frequency-responsive network driven by one or more active drivers. The typical
driver typically is an opamp or one of its cousins. Often the circuit can then be created in an integratedcircuit form in which the only frequency-dependent components are capacitances.
The integrated circuit active filter is also appropriate to RF (radio frequencies) as a means to either avoid
or to accommodate the parasitic wiring capacitances and inductances.
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A frequency profile is generically defined by a transfer function T(s) = |T(s)| ej(s) in the s-domain, for
which s = j. And as identified by chapter 4, the graphical representations of |T(s)| and (s) are called
Bode magnitude and Bode phase plots, respectively, after their originator, Hendrik Bode (1905-1982).
The order of the profile T(s) is defined by the number of frequency-dependent components. All profiles
will have a numerator N(s) and a denominator D(s)
T ( s) 
N ( s)
D( s )
(17.1-1)
for which the numerator zeros are called zeros and the denominator zeros are called poles. Poles are
associated with roll-off of the frequency profile. Zeros will have the effect of ‘roll-upward’. All transfer
functions have the same number of zeros as they do poles. The effects of poles are almost always visible.
The effects of zeros may be less visible since many zeros will fall either at zero or at infinity.
A great deal of attention has been given to the art of frequency profiling, for which placement of poles
and zeros is of critical importance to the character of the profile. Therefore many profile options will be
identified in terms of names of their benefactor or in terms of mathematical descriptors. We will not
attempt to survey all of the different types except to say that there are enough to satisfy most of the
discriminating engineering requirements. Usually the different types will exist in tabulated forms.
For simplicity and conservation of context the table values for profiles of interest can be assumed and
then we will load and go, provided we know how to rescale and convert to the frequencies of interest and
have enough perspective to make a judicious choice.
Bilinear circuits: The bilinear circuits are the simplest category of frequency profile. The transfer
function T(s) is a ratio of two linear functions, otherwise called a bilinear form. The general form for the
bilinear function is
T (s)  K
sz
s p
 K'
1 s z
1 s p
(17.1-2)
Table 17.1-2 shows the magnitude response for the two single-time-constant types of bilinear response
(low-pass and high-pass) and their accompanying phase responses. The roll-off slope is of the form of
–20dB/decade (which is the same as a ratio of -1/1). The roll-off slope on the log-log plot is linear, since
for for  > 10p, the magnitude of the denominator becomes approximately linear, i.e.
D( s)  1   p    p
2
as  >> p
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Circuits, Devices, Networks, and Microelectronics
Table 17.1-1(a). Bode magnitude and phase response for low-pass profile.
Table 17.1-1(b). Bode magnitude and phase response for high-pass profile.
Bode magnitude plots are usually represented as log-log scales with the frequency axis of the form of a
decade scale and the amplitude axis of the form of a magnitude scale in dB. The zeros are usually at
either infinity or zero (which is typical) and beyond the range of a finite Bode plot.
For a more emphatic roll-off, higher-order transfer functions are required. Assuming appropriate
placement of poles and zeros roll-off becomes as much as ~ n x (-20dB/dec), where n = number of poles,
The case should be stated for more than the |T(s)| roll-off in terms of the low-pass profile (Table 24-1.1),
Low-pass is an inevitable fact of all electronic circuits. The higher the frequency the more difficulty the
signal has of passing through a network and the more readily it is shunted to ground. And so
T (s  )  0 .
Biquadratic circuits: The simplest categories of frequency profiles are (1) first-order forms and (2)
second-order forms. The second-order are profiles are called biquadratic forms. The biquadratic form
yields a complete set of pass-band profiles as represented by Table 17.1-2. The general form of the
biquadratic function is
T ( s)  K
n 2 s 2  n1 s  n0
s 2  s  0 Q   02
 K'
1 s a  s2 b
1  s  0 Q  s 2  02
(17.1-3)
Table 17.1-2 shows the types of magnitude response for different numerator functions. For convenience
and definition of the key profile parameters, the denominator function is always of the form
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D(s)  s 2  s  0 Q   02
D(s)  1  s  0 Q  s 2  02
or
(17.1-4)
Equation (17.1-4) is the preferred form for the quadratic denominator since it relates to a characteristic
(resonance) frequency 0 = 2f0 and a ‘quality factor’ Q. The band-pass function, for which n2 = 0 and
n0 = 0, will have peak magnitude at f0 and will be symmetric about f0 on the decade frequency scale.
Low-pass and high-pass quadratic functions will also show a profile peak if Q > 1
2 , although it will
not be very pronounced until Q > 2.
Type
(Amplitude normalized) transfer
function
(a) Band-pass
T (s) 
s 0 Q
s  s  0 Q   02
2
(Q = 4 represented)
(b) Low-pass
T (s) 
 02
s 2  s  0 Q   02
(Q = 4 represented)
(c) High-pass
T (s) 
s2
s 2  s  0 Q   02
(Q = 4 represented)
(d) Band-stop
T (s) 
s 2  s  0 Q '   02
s 2  s  0 Q   02
(Q = 4 represented)
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(e) All-pass
T (s) 
s 2  s  0 Q   02
s 2  s  0 Q   02
(Q = 4 represented)
Table 17.1-2. The equations shown are normalized to |T(max)| = 1.0 for the bandpass, |T(0)| = 1
for the low pass and |T(∞)| = 1 for the high-pass. Biquadratic profiles usually serve as benchmarks for
their higher-order cousins.
The profile for the profile 17.1-2(e) (all-pass) bears some explanation. It is more correctly identified as a
phase-shift circuit for which the phase varies as a function of frequency but the amplitude does not. Since
it is of second-order it will shift phase by = -2 x 90o = -180o across the span 0 <  < ∞ .
The normalized form of the band-pass function, for which n1 = 0/Q , is a means to identify the reason
why Q is called the quality factor of the profile. Since the band-pass function is a means to select and
pass a given frequency it is appropriate to identify the bandwidth of the profile  = 2 – 1 , which is
defined in terms of the 3-dB levels ( T  1
2 ), a.k.a. half-power levels as indicated by figure 17.1-3.
Figure 17.1-3. (Normalized) biquadratic band-pass function
Analysis of the normalized band-pass equation (given in Table 17.1-2(a) at magnitude levels for which
T 1
2 , gives a quartic equation with four solutions:
 
0
2Q

0
2Q
1  4Q 2
(17.1-5)
Of these solutions the only ones that are greater than zero are
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1  
for which
0
2Q

0
2Q
1  4Q 2
   2  1 
and
2  
0
2Q

0
2Q
1  4Q 2
0
(17.1-6a)
Q
which represents the selectivity characteristic of the resonance peak, i.e.
Q
0


f0
f
(17.1-6b)
which is why Q is called the ‘quality factor.’ The mathematics is detailed in Chapter 4 and equation
(17.1-6b) is the same as equation (4.5-11). Otherwise emphasis should be made that Q is a magnitude
factor for the low-pass profile resonance peak since
Q
T (   0 )
(17.1-7)
T (  0)
and which was examined more carefully in chapter 16 as well as in chapter 4.
As represented by figure 17.1-2 the low-pass and high-pass profiles of primary interest are (1) Q = 1
2,
for which the profile is maximally flat and (2) Q > 4 , for which the resonance peak begins to dominate.
Figure 17.1-2: Low-pass profile for several values of Q, i.e. Q = {0.5, 0.707, 2, 4}. The resonance peak
is approximately at f0 for Q > 4.
There is more to the story, however, when we extend the reach to specific applications. They are
categorized according to their properties as follows:
(1) Butterworth profiles
(2) Chebyshev profiles
= maximally flat in the pass-band
= maximum roll off at the pass-band corner
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(3) Bessel (Thomson)
= maximum linearity in the phase response and minimum
overshoot for impulse signals
There are others which may accept ripple in both pass-band and stop-band in order to achieve better
quality response for one of the above categories. But each of the three categories cited above the
biquadratic response may be related to Q as follows:
(1) Butterworth:
Q=1
(2) Chebyshev:
(3) Bessel:
Q = 1.0
Q = 0.5
2
(3dB ripple )
The Chebyshev profile is a little more artistic than suggested by these simple relationships, since it may
be either of three types. Type (1) has ripple in the pass-band, type (2) has ripple in the stop-band and
type (3) (a.k.a. elliptic or Cauer response) has ripple in both pass-band and stop-band. The Chebychev
name is a consequence of the mathematics, for which the poles and zeros are defined by Chebyshev
polynomials. In the s-plane its poles will pattern in the form of an ellipse.
The ripple for the Chebyshev profile may also be of different relative magnitudes. As might be expected
for the biquadratic form, a Q > 1.0 is also a citation for more ripple in the pass band and the benefit of a
greater roll-off at the break point. The circuit designer then takes up a role as an artist in mathematics
and circuit electronics.
17.2 NORMALIZED FORMS and FREQUENCY RESCALING
Many circuit topologies have been analyzed, addressed, profiled and tabulated by generations of electrical
engineering slaves. Tabulated forms are normalized to a characteristic frequency 0 =1.0 r/s or to a
feature frequency 1 =1.0 r/s. The normalized form of the topology is typically the one to which
rescaling techniques of different modality are applied.
Frequency rescaling is the analytical process of rescaling the components of a circuit topology to one that
will yield an exact copy of the profile at another frequency. Frequency rescaling can also be used to
transform the circuit into an equivalent topology with the same profile, in which case it is called a
frequency transformation.
Circuit profiles is defined in terms of a set of zeros in the numerator N(s) terms and another set of zeros in
the denominator D(s) defined as poles because of their effect on the profile. Poles and zeros will each
have a relationship to a time constant. And if the time constants are rescaled to a new s’ =j’ plane then
they will have the exact same profile if  x1 = ' x2 . Frequency response is defined by time constants
(duh!). For the simplest case, for which all of the frequency dependent components in a circuit are
capacitances then 1 = R1C1 and 2 = R2C2, and the s and s’ planes will relate to one another by
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Circuits, Devices, Networks, and Microelectronics
R1C1   ' R2 C2
(17.2-1)
where we are identifying that s = j and s’ = j’.
So if we do not change resistance, i.e. R1 = R2, then (17.2-1) reduces to
C1   ' C 2
(17.2-2a)
and we can shift the profile from the s to the s’ plane by merely rescaling the capacitance according to
C2 

C1
'
(17.2-2b)
For example if the s’ domain is a rescale of 1000 s, for which ’ = 1000, then the capacitances must be
1000 times smaller. A 1.0mF capacitance would become a 1.0F capacitance.
If we do not change capacitance, i.e. C1 = C2, then
R1   ' R2
(17.2-3a)
and we can shift the profile from the s to the s’ plane by rescaling the resistance according to
R2 

R1
'
(17.2-3b)
For example if the s’ domain is 1000s, (for which ’ = 1000, the resistances must be 1000 times
smaller. (And for ’ = 1000 the circuit is 1000 times faster since RC is 1000 times smaller).
Actually we would most likely change both R and C, in a process called RC rescaling. This process is
accomplished by a two-step procedure. (1) We keep R unchanged, i.e. R1 = R2, and rescale the
capacitances to a new frequency M, i.e.
C1   M C2
for which
and for which
M 
C1

C2
(17.2-4a)
R1C1   M R1C2
For step (2) we hold C2 constant and rescale the resistances, for which
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 M R1C2   ' R3C2
R3 
for which
M
R1
'
(17.2-4b)
Now both R and C are changed and the profile is translated into the s’ plane to an identical profile which
usually is at a higher frequency.
The process in a different order, and is indicated by figure 17.2-1. The most likely option is to elect a
final capacitance value C2 and then let the resistance follow. Equations (17.2-4a) and (17.2-4b) represent
the conventional RC rescaling algorithm:
M 
and
Cinit
init
C final
R final 
(17.2-5a)
M
Rinit
 final
(17.2-5b)
The frequency M in equations (17.2-5) is defined as the rescaling frequency. The init and final are
initial and final frequencies, respectively. The mathematics is further simplified if the initial frequency
init = 1.0 r/s, i.e. that associated with a normalized profile.
The same process can be accomplished with inductances except by means of the time constant   L / R
for which the equivalent to equation (17.2-1) is then
 L1 R1   ' L2 R2
(17.2-6)
but it is not common to rescale according to inductances.
A diagrammatic representation of RC rescaling using equations (17.2-5) is shown by figure 17.2-1.
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Circuits, Devices, Networks, and Microelectronics
Figure 17.2-1 RC rescaling steps as represented by two rescaling paths. The upper path is
usually the path of choice and matches equations (17.2-5).
There are several good candidates for RC rescaling that we will check out in later sections of this chapter.
For topologies that include both capacitances and inductances the rescaling process is not quite as simple.
For example, the RLC biquadratic circuits will have
 02 
1
LC

R 1
1
1



L RC  L  C
(17.2-8)
So the resistances cannot be rescaled by the simple time-constant method applied to L and C separately.
A different paths must be followed which includes an impedance magnitude rescaling. The rescaling
algorithm will then be of the form
R2  k M R1
C2  C1 k M k f
L2  L1 k M k f
(17.2-9(a)


(17.2-9(b)
(17.2-9(c)
Quadratic rescaling must then be accomplished by means of either (1) a component translation which
retains the simplicity of equations (17.2-1) or (17.2-7) or (2) paths like those represented by equations
(17.2-9). Since the translation is essentially an equivalent circuit form it invites usage of amplifier-driven
components and the reconfiguration of the topologies into an active-filter form.
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Circuits, Devices, Networks, and Microelectronics
17.3 BIQUADRATIC CIRCUIT TOPOLOGIES
There are a number of RC topologies that can be invited to the biquadratic table. Almost all of them
employ one or more gain elements, typically of the form of an ideal opamp, which certainly has its
caveats if it turns real. But otherwise the opamp is a reasonable drive element for applications in which
biquad profiles (table 17.1-2) are elected.
For example the Sallen-Key circuit topology shown by figure 17.3-1 is a benchmark standard. It is
simple, requires only one amplifier component, and the Q of the circuit is separately tunable relative to
the characteristic frequency 0.
Figure 17.3-1 Sallen-Key single-amplifier biquad.
Nodal analysis at v1 and v+ gives
v1 (G1  G2  sC1 )  vo sC1  v s G1  v G2  0
v (G2  sC 2 )  v1G2  0
Using vo  Kv  , where K 1  Rb Ra gives
T ( s) 
G G
vo
 K  1 2
vs
 C1C 2



 2
 G2 1  K )  G1  G2

s  s
C2
C1


 G1G2 
 

 C1C 2 
(17.3-1)
Typically we let C1 = C2 = C and G1 = G2 = G for which
T ( s) 
v0
K 02
 2
v s s  s 0 3  K    02
(17.3-2)
where  0  G C  1 / RC . Note that (17.3-2) is of the low-pass form, with Q = 1/(3 – K). So Q can be
tuned by means of an adjustment of the ration of Rb/Ra according to
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Q  1 2  RB / R A 
(17.3-3)
Since the ratio G2/C2 is consistent throughout equation (17.3-1) we may taper the Sallen-Key biquad by
selecting C2 = aC1 = aC and G2 = aG1 = aG . This modification does not change the characteristic
frequency 0, but will change the form of the expression for quality factor to the form
Q  1 1  a  Rb / Ra 
(17.3-4)
In particular, if the gain element is set up for K = 2 (for which Rb/Ra = 1) as shown by figure 17.3.2 and Q
is defined by tapering factor a, then the circuit simplifies to Q  1 a
Figure 17.3-2. Design-B of the Sallen-Key for tunable Q.
Note that we have set the design in normalized form (0 = 1.0r/s), to which we can apply the RC rescaling
algorithm (17.2-5) as shown by figure 17.3-3. In this realization you might note that the parameter
rescaling forms have made usage of the approximation 1 2  0.16 .
Figure 17.3-3. Design B of the Sallen-Key with RC-rescaling to10kHz and Q stepped through
several values. Also take note that the zero-frequency gain K = 2.0.
Another form of the Sallen-Key topology is the Saraga design shown by figure 17.3-4.
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Figure 17.3-4. Saraga low-sensitivity design for the Sallen-Key single-amplifier biquad with 0
normalized to 1.0 r/s. Note that for the Saraga design, R1 = 1/Q. The topology may then be rescaled to
the frequency domain of interest by RC rescaling.
We can use any number of active components in a circuit, and sometimes it is of practical benefit to do so
as represented by the two-integrator loop of figure 17.3-5. It has three output points, each of which
differs by the factor –0/s.
Figure 17.3-5. Two-integrator loop.
The factor –0/s is generated by the ‘Miller’ integrator
which is a straightforward and simple circuit. The two-integrator loop consists of two Miller integators
and one inverter-summing circuit as shown by figure 17.3-5, for which the output of the summing circuit
will be
vo 
2
1  0 

vo  20 vo  n2 v I
Q s 
s
(17.3-5)
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Circuits, Devices, Networks, and Microelectronics
Collecting like terms and resolving the transfer function from vI to vo yields high-pass transfer function
THP ( s) 
vo
n2 s 2
 2
v I s  s 0 / Q   02
(17.3-6)
From this equation and the relationship between stages
n 2 0 s
v1 vo   0 
  
 2
vI vI  s 
s  s 0 / Q   02
 TBP (s)
(17.3-7a)
band-pass
n2 02
v 2 vo   0 

 
 2
v I v  s 
s  s 0 / Q   02
 TLP (s)
(17.3-7b)
low-pass
and so two-integrator loops are also called state-variable filters since they will provide the three
basic functions of low-pass, high-pass, and band-pass.
The two-integrator loop most often deployed is the Tow-Thomas, or ring-of-three topology shown by
figure 17.3-6. One of the integrators is a lossy integrator, and it is this choice that gives the circuit its
operational benefit.
Figure 17.3-6. Tow-Thomas (ring-of-three) two integrator topology.
Nodal analysis at the input of opamp U1 gives
0  v s G3  v2 G4  v1 G1  sC1 
(17.3-8a)
and since U3 is a unity-gain inverter and U2 is a Miller integrator then
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Circuits, Devices, Networks, and Microelectronics
v2  
G2
v1
sC 2
(17.3-8b)
Solving for v1 we get
 s G3 C1
v1
 2
v s s  s G1 C1  G2 G4 C1C 2
 TBP (s)
(17.3-9)
band-pass
 TLP (s)
(17.3-10)
low-pass
and from equation (17.3-8b) we get
 G2 G3 C1C 2
v2
 2
v s s  s G1 C1  G2 G4 C1C 2
Typically we would (1) choose C1 = C2 = C and R2 = R4 = R, for which  0  1 / RC . Then (2) choose Q =
R1/R. Then (3) choose amplitude k = R1/R3. Using this tuning algorithm, the profile parameters 0, Q,
and k are separately adjusted in the order (1) thru (3).
The tuning simplicity of the Tow-Thomas topology is even more evident when it is put in normalized
form, for which we would let C1 = C2 = 1.0F and R2 = R4 = 1.0.Then R1 = Q and R3 = Q/k, and the
topology is of the form shown by figure 17.3-7, which also includes an extra summing amplifier for the
generation of the band-stop function.
Figure 17.3-7. Tow-Thomas topology in normalized form configured with extra summing
amplifier U4 for the generation of band-stop function.
In normalized form equations (17.3-9) and (17.3-10) will be
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Circuits, Devices, Networks, and Microelectronics
v1
 sk / Q
 2
vs s  s / Q  1
 TBP (s)
(17.3-11a)
band-pass
v2
 k /Q

vs s 2  s / Q  1
 TLP (s)
(17.3-11b)
low-pass
The band-stop output is provided by summing opamp U4 for which we have an extra output
node v4 = vs + v2 and for which
v4 vs v2


vs vs vs
1
 sk / Q
2
s  s / Q 1

s 2  s(1  k ) / Q  1
s2  s / Q 1
(17.3-12)
Equation 17.3-12 is of the form of a band-stop function. Its behavior for a simulation in which k = 0.4
and Q = 4 is shown by figure 17.3-8. If we choose values such that k = 1, then the band-stop function
becomes a band-reject form.
Figure 17.3-8. Tow-Thomas simulation using tuning algorithm and normalized form as rescaled to 1.0kHz.
Low-pass and band-stop outputs are shown. Peak amplitude of the low-pass is 0.4 and the minimum value
of the band-stop is 0.6. Note that the Tow-Thomas topology acts as a filter and not as a resonator (i.e. no
outputs have greater magnitude than vs).
If we choose k = 2, then a special case biquadratic function occurs at v4 with transfer function
T (s) 
s2  s / Q 1
s2  s / Q 1
(17.3-13)
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This function is of the form of an all-pass function and serves only to produce a phase shift. The phase
shift = -90o at the characteristic frequency f0 = 20.
17.4 BIQUADRATIC FILTER FUNCTIONS
If we factor the biquadratic denominator D(s)  s 2  s  0 Q   02 the roots are
s
0 
1  j 4Q 2  1 
2Q 
and

s
0 
1  j 4Q 2  1 
2Q 

(17.4-1)
for which it is evident that the roots are complex when Q > 0.5.
The s-plane is complex, and these roots are located in the left (negative) half of the s-plane. As the
parameters 0 and Q, are varied a root-locus plot of these roots will result, as represented by figure 17.11.
Figure 17.4-1. Root-locus plot for the poles of a biquadratic function. This figure is a
modified copy of Figure 16.5-1 with more emphasis on the role of Q.
In particular, we can see that the root-locus plot for 0 = constant is a circle with radius 0. If we
determine the magnitude of either of the roots identified by equation 17.4-1, we find that
s
0
2Q


 1  4Q 2  1
and that different values of Q
and for which
 0
(17.4-2a)
correspond to different values of the angle , as shown by figure 17.4-1,
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cos  
0
(17.4-2b)
2Q
We often use the roots and root loci to identify different profiles according to location of the poles and
zeros. For example low-pass Butterworth profile with a roll-off of n x (-20dB/decade), is of the form
H (s) 
2
and will have roots at
k
(17.4-3a)
1  s / j 0 
2n
s   0 e j ( 2k ) / 2n   0 e jk
(17.4-3b)
where k = 0, 1, 2 … n, and for which the equation looks like the same form of entertainment you used to
have with the mathematics of complex variables.
If n = 2, then the two roots that fall in the left half plane are at k = (3/4) and (5/4), both of which relate
to the angle = 45o, for which Q = 1
2 . So the maximally flat profile is the Butterworth profile by
another name.
If n = 4, then the four roots that fall in the left half plane are at k = (5/8), (7/8), (9/8), (11/8). These
correspond to two values of Q = 1/(2cos(22.5o)) = 0.541, and Q = 1/(2cos(67.5o)) =1.307, and can be
accomplished by a couple of Sallen-Key topologies in cascade, as shown by figure 17.4-2.
Figure 17.4-2 Use of Sallen-key topologies in cascade to generate 4th-order (maximally-flat)
Butterworth response. Take note that the roll-off slope is –80dB/decade = same as 4 x (–20dB/dec).
The root-locus plots are also of advantage to the use special mathematical functions to identify pole and
zero distributions that will effect a sharper cut-off at the break frequency fc of the profile. A fairly popular
class of sharp cutoff functions is based on Chebyshev polynomials according to transfer function
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H ( s) 
k
1   cos C n ( /  c ) 
2
2

(17.4-4)
2

where the function C n ( x)  n cos1  /  c is a Chebyshev polynomial, and is probably familiar only to
them who is well-versed in special mathematical functions. The factor  is called the ripple factor and
defines the ripple in the pass-band. As represented by figure 17.4-3, which shows a 3dB ripple, the ripple
factor  = 1. Cutoff frequency is referenced to f c  2   c .
Figure 17.4-3. 4th-order Chebyshev profile with  = 1.
And if we indulge in the mathematics we would find that the choice of 01 = 0.951c , Q1 = 5.59 and 02
= 0.443c , Q2 = 1.075 for two Sallen-Key biquads in series will result in a -3dB Chebyshev 4th-order
profile as shown by figure 17.4-4.
Figure 17.4-3. Sallen-Key implementation of 4th-order Chebyshev profile with  = 1. The accompanying
trace is that for the 4th-order Butterworth profile.
A source of biquadratic topologies with names and provenance is located under the Texas Instruments
Split-supply Analog Filter expert. It not only offers topologies for the standard biquadratic forms but
deploys them according to the more common filter properties (i.e. Butterworth, Bessel and Chebyshev).
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17.5 DOUBLY-TERMINATED RLC LADDERS and TRANSFORMATIONS
The section on biquadratic forms is a small slice of the collection of frequency profiles that are
standardized. For applications and realizations beyond the biquadratic there are two means by which we
may glean higher-order profiles: (1) poles and zeros defined by polynomials (e.g. Butterworth,
Chebyshev, Bessel) and (2) RLC tables. The polynomials are probably more mathematically satisfying
but the tables are more convenient. It is not uncommon to define the rescaling and transformation
algorithms in terms of a doubly-terminated RLC topology, such as is represented by figure 17.5-1. A
subset of the tables are shown by figure 17.5-2, and credit for them belongs to one of signature resources
for analog filter design (M.E.Van Valkenburgh, Analog Filter Design, HRW, 1982. Unfortunately much
of its content is not in softcopy and so the tables have to be included in hardcopy.
Figure 17.5-1. Doubly-terminated RLC ladder topologies (equivalent forms)
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Figure 17.5-2. Doubly-terminated RLC ladder tables (this is only a subset). The strip index at the top
beginning with L1 is associated with the RLC ladder above the tables and the strip index at the bottom
beginning with C1 is associated with the RLC ladder below the tables.
Direct simulations using the RLC ladder topologies and table values will yield a normalized low-pass
profile normalized. The categories (or polynomial types) are listed above the tables. All of the values are
in Ohms, Farads, or Henrys, depending on the component. The rest of the story is that the tables are the
touchstone for a profile of any of the generic types, low-pass, high-pass, band-pass, band-stop. Profiles
are realized by means of (1) transformation and (2) frequency rescaling.
The mathematics of making a transformation is entertaining, usually with the idea that a transfer function
T(s) is a function of impedance/admittance ratios and the fact that if we transform them in the same way,
the ratio will remain the same. Add a translation to this context and a low-pass can be translated into an
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equivalent form with two shoulders instead of one, if we regard the low-pass as a one-shoulder profile
centered about zero. All that is needed is to shift the zero and the other shoulder emerges.
But the greater benefit of the process is that an RLC circuit can be translated into a form that employs
impedance-converter circuits with opamps to eliminate the bulky inductances. And that takes the process
into a form that can be micro-miniaturized since a VLSI chip can literally contain dozens of opamps.
The active-filter context (using opamps) is then resolved as an R-C circuit, which can be translated to
whatever frequency characteristic is needed using the algorithm of figure 17.2-1 or its equations (17.2-5).
And it surely makes life more simple than the mathematical divinations.
For example take a look at a transformation called the RLC:CRD transformation, so-called because the R
is recast as a C, the L is recast as an R and the C is recast as an impedance-conversion construct element
(= D) called a frequency-dependent negative resistance (FDNR). The concept is illustrated by Figure
17.5-3.
Figure 17.5-3. RLC:CRD transformation
The technique is also called the Bruton transformation and is primarily a mathematical manipulation, as
represented by equation (17.5-2)
Figure 17.5-3(a) has transfer function
T ( s) 
v2
1 sC

v1 R  sL  1 sC
(17.5-1)
And figure 17.5-3(b) has transfer function
T ( s) 
v2
1 s 2C

v1 R s  L  1 s 2 C

1 s2D
1 sC '  R '  1 s 2 D
(17.5-2)
Where (17.5-2) is (17.5-1) with both numerator and denominator multiplied by (1/s).
And the change of labeling does not bother the computation.
But the new component is an opamp construct derived from the Fleige topology, as represented by figure
17.5-3.
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Figure 17.5-3. (a) Generalized impedance converter (GIC) topology and (b) translation into FDNR (= D
component).
Analysis of figure 17.5-3(a) is a matter of nodal analysis at each of the three nodes labeled v0. They are
all virtually connected because of the nullator action of the opamps. Eliminating v1 and v2 from the
resulting three equations gives
Y1Y3Y5  Y2Y4Y6  0
(17.5-2)
Which can be solved for Yin = -Y1 as
Yin 
Y2Y4Y6
Y3Y5
(17.5-3)
With the component choices indicted by figure 17.5-3(b) and making the choice R3 = R4 we have the
desired FDNR behavior
Yin  s 2 C2 C6 R5  s 2 D
(17.5-4)
where the magnitude of D is then  C 2 C6 R5 .
The process is just a bit of mathematics. But the mathematics more easily accomplished if developed on
a simulation platform as represented in previous section of this chapter. The process is illustrated by
figures 17.5-4 and 17.5-5:
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Figure 17.5-4(a) Doubly-terminated RLC topology for a 5th-order Chebyshev with 1.0dB ripple in the
pass-band. Note that the parameter values are the number values from Chebyshev (C) of figure 17.5-2.
Figure 17.5-4(b) Simulation results. Note that the breakpoint is at f0 = (1/2) x 1.0 rad/sec = 160 mHz.
So this profile is the normalized form.
Figure 17.5-5 Same as figure 17.5-4(a) except implemented in CRD form. The L’s are replaced by R’s,
the R’s are replaced by C’s, and the capacitances are replaced by FDNR’s.
The parameterization in figure 17.5-5(a) is doing all of the mathematics grunt work, to include the
frequency rescaling. If you look at the very bottom of the figure you will see that the capacitance has
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Circuits, Devices, Networks, and Microelectronics
been chosen to be .01 F. And then the rescaling algorithm according to equations (17.2-5) is located in
the parameter list in the lower left corner. Notice that the 1.0r/s normalized frequency is translated into
0.16 since the simulation is always in Hz. The units are omitted since it is the mathematics that carries
the process along. The scaling factor shown as kR is applied to all resistances, exactly as prescribed by
equation (17.2-5(b)).
The diagram is a little busy, but you should take note of a few simplifications that are used and are not
uncommon.
Due to the RLC:CRD transformation the capacitances relate to the R = 1.0 termination resistances, and
hence are normalized to 1.0F. The other capacitances, which belong to the FDNR, are dealer’s choice.
So they might as well be elected to be the same, i.e. 1.0F capacitances. This choice also simplifies the
(FDNR) value for D, which by equation (17.5-4) gives
D = C2 C6 R5 = 1.0 x 1.0 x R5
= R5
(17.5-5)
So it is not uncommon to deploy the CRD representation in the form of an equal capacitance realization.
And that lets the rescaling be a relatively simple RC rescaling process, as indicated by the parametric
usage in the lower right corner of figure 17.5-5(a).
For the final frequency (labeled as fC in the figure), the rescaled CRD realization gives the frequency
profile result as shown by figure 17.5-5(b).
Figure 17.5-5(b) Simulation results. Both normalized profile and rescaled profile at fC = 10 kHz are
shown as a comparison to confirm that the transformation/rescaling will realize the same profile but at a
different frequency.
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Circuits, Devices, Networks, and Microelectronics
There are a few extra wrinkles that are necessary which the mathematics does not represent. The
capacitances that replaced the terminations (at each end) leave the circuit components floating relative to
ground, and therefore must be accompanied by leakage resistances (shown as 999M values). If this is
not done the circuit simulator willabort and give an error this effect. The values of the leakage resistances
are arbitrary, but large.
The RLC-CRD process is a generic and common choice in active-filter design. Others of note are ‘LeapFrog’ design, which use a clever process to transform the RLC ladder to an equivalent form that will
accomplish to a ‘low-pass to band-pass’ (LP:BP) transformation. But in the interest of keeping the topic
area of frequency profiling reasonably compact, LeapFrog design will be omitted except for name
acknowledgement. Regrets.
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Circuits, Devices, Networks, and Microelectronics
PORTFOLIO and SUMMARY
RC frequency rescaling
M 
Cinit
init
C final
R final 
and
M
Rinit
 final
Sallen-Key biquad (lowpass ) = tuneable Q
0 = 1/RC
Q  1 1  a  R f Rx 
where a = tapering factor
Tow-Thomas biquad = biquad states for (1) low-pass, (2) band-pass, (3) band-stop, (4) all-pass
0 = 1/RC = 1.0 r/s
normalized
Normalized values:
C1 = C2 = 1.0F
R2 = R4 = 1.0
R1 = Q
R3 = Q/k
TBS (s) 
v S  v1 v4 s 2  k / Q  1


vs
vs s 2  s / Q  1
TBP (s) 
v1
 sk / Q
 2
vs s  s / Q  1
TLP (s) 
v2
 k /Q
 2
vs s  s / Q  1
TAP(s) = TBS(s) when k = 1
Butterworth profiles = maximally flat in the pass-band
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Chebyshev profiles = maximum rolloff at the pass-band corner
Bessel (Thomson) = maximum linearity in the phase response, minimum overshoot for impulse signals,
minimum delay of signals
Simulations: 5th-order low-pass. Profiles and pulse response
Chebyshev (1.0dB)
Butterworth
Bessel
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